Randomness-Efficient Sampling within NC¹

Abstract.

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in uniform AC ⁰[⊕]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC ¹. For example, we obtain the following results:

• Randomness-efficient error-reduction for uniform probabilistic NC ¹, TC ⁰, AC ⁰[⊕] and AC ⁰:

  Any function computable by uniform probabilistic circuits with error 1/3 using r random bits is computable by circuits of the same type with error δ using r + O(log(1/δ)) random bits.

• An optimal bitwise ϵ-biased generator in AC ⁰[⊕]: There exists a 1/2^Ω(n)-biased generator G : {0, 1}^O(n) → {0, 1}²ⁿ for which poly(n)-size uniform AC ⁰[⊕] circuits can compute G(s) _i given (s, i) ∈ {0, 1}^O(n)  ×  {0, 1}ⁿ. This resolves question raised by Gutfreund and Viola (Random 2004).

• uniform BP · AC ⁰ ⊆ uniform AC ⁰/O(n).

Our sampler is based on the zig-zag graph product of Reingold, Vadhan & Wigderson (Annals of Math 2002) and as part of our analysis we givean elementary proof of a generalization of Gillman’s Chernoff Bound for Expander Walks (SIAM Journal on Computing 1998).